representations with centralizer stable under conjugate transpose Let $\rho:G\to GL_n(\mathbb{C})$ be a finite-dimensional representation of a finite group $G$ over $\mathbb{C}$, and $C_\rho\subset M_n(\mathbb{C})$ its centralizer, i.e. $m\in C$ iff $m$ commutes with each $\rho(g)$, $g\in G$.
In some cases, e.g. if $\rho$ is irreducible, or $\rho$ is unitary, $C_\rho$ is closed under conjugate transpose $*$, where $m^*:=\overline{m}^\top$.
What are reducible non-unitary $\rho$ for which $C_\rho$ is closed under conjugate transpose?
 A: I do not have a complete answer, but it might be helpful anyway.
Since every representation of a finite group is unitarisable, there is an invertible operator $T$ and a unitary representation $\sigma$ such that $\rho(g) = T \sigma(g) T^{-1}$. If $C_{\sigma}$ denotes the centraliser (or commutant) of $\sigma$, then $C_{\rho} = T C_{\sigma} T^{-1}$. Now $C_{\sigma}$ may be an arbitrary unital, $\ast$-subalgebra. The condition $C_{\rho}^{\ast}=C_{\rho}$ is equivalent to $T^{\ast}T C_{\sigma} (T^{\ast}T)^{-1} = C_{\sigma}$. Since $T^{\ast}T$ is positive, there exists a unitary $U$ such that $T^{\ast}T = UD U^{\ast}$, where $D$ is a diagonal matrix. The condition we get now is $U^{\ast}C_{\sigma} U = D U^{\ast}C_{\sigma} U D^{-1}$. Now we would like to force $U^{\ast} C_{\sigma} U$ be a subalgebra of block diagonal matrices, where the structure of blocks is determined by the representation. Note that any unital, $\ast$-subalgebra of $M_n$ is conjugate by a unitary to such a thing, so we may achieve that by replacing $T$ by a slightly different operator. Namely, fix a unitary $V$ such that $VC_{\sigma} V^{\ast} = B$, where $B$ is a block diagonal subalgebra. If we replace $T$ by $TUV$ (i.e. we consider new $\sigma'(g):= (UV)^{\ast} \sigma(g) UV$), then we get what we want. And in the block diagonal case the conjugation by a diagonal matrix preserves the block structure, so everything is fine.
So, it seems that the main point is to find an equivalent unitary representation and then tweak it a little bit, so that the centraliser is of a particularly nice form. The procedure in this answer does not exhaust all the possibilities but at least it shows the following: if $\sigma$ is a unitary representation and $V$ is a unitary such that $VC_{\sigma}V^{\ast}$ is block diagonal then any $T=UDV$, with $D$ diagonal and $U$ unitary will provide an example of a representation $\rho = T\sigma T^{-1}$, whose centraliser is self-adjoint.
